Problem: Let $x$ be a real number.  Consider the following five statements:

$0 < x^2 < 1$
$x^2 > 1$
$-1 < x < 0$
$0 < x < 1$
$0 < x - x^2 < 1$

What is the maximum number of these statements that can be true for any value of $x$?
One the first two statements, at most one of them can be true ($x^2$ cannot be both less than 1 and greater than 1).  Of the next two statements, at most one of them can be true ($x$ cannot be both less than 0 and greater than 0).  Hence, at most three statements can be true.

For $0 < x < 1,$ the first, fourth, and fifth statements are true, so the maximum number of statements that can be true is $\boxed{3}.$